Information structures and their cohomology

Juan Pablo Vigneaux

We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings. Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement. We extend Baudot and Bennequin's definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature. Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions. The only 1-cocycles are Shannon entropy or Tsallis alpha-entropy, depending on the value of the parameter.

Keywords: information cohomology, entropy, nonextensive statistics, information structures, sheaves, topos

2020 MSC: 55N35, 94A15, 39B05, 60A99

Theory and Applications of Categories, Vol. 35, 2020, No. 38, pp 1476-1529.

Published 2020-08-24.

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